Dissolve 2 tablespoons salt in 1 quart cold water. Submerge chicken in brine, cover, and refrigerate for 30 minutes. While chicken is brining, combine oil, parsley, lemon zest and juice, honey, oregano, and½ teaspoons pepper in medium bowl. Transfer¼ cups oil mixture to large bowl and set aside to toss with cooked chicken.

Remove chicken from brine and pat dry with paper towels. Toss chicken with remaining oil mixture. Thread 4 pieces of bell pepper, concave side up, onto one 12-inch metal skewer. Thread one-quarter of chicken onto skewer. Thread 2 chunks of onion onto skewer, and place on plate. Repeat skewering remaining chicken and vegetables on 3 more skewers. Lightly moisten 2 pita breads with water. Sandwich 2 unmoistened pita breads between moistened pita breads and wrap stack tightly in lightly greased heavy-duty aluminum foil.

Turn all burners to high, cover, and heat grill until hot, about 15 minutes. Leave primary burner on high and turn off other burner(s).

Clean and oil cooking grate. Place skewers on hotter side of grill and cook, turning occasionally, until chicken and vegetables are well browned on all sides and chicken registers 160°, 15 to 20 minutes. Using fork, push chicken and vegetables off skewers into bowl of reserved oil mixture. Stir gently, breaking up onion chunks; cover with foil and let sit for 5 minutes.

Meanwhile, place packet of pitas on cooler side of grill. Flip occasionally to heat, about 5 minutes.

Lay each warm pita on 12-inch square of foil. Spread each pita with 2 tablespoons tzatziki (see recipe in "sauces"). Place one-quarter of chicken and vegetables in middle of each pita. Roll into cylindrical shape and serve.